Degree conditions on distance 2 vertices that imply k-ordered Hamiltonian

  • Authors:

  • Ruijuan Li;Shengjia Li;Yubao Guo

  • Affiliations:

  • School of Mathematical Science, Shanxi University, 030006 Taiyuan, PR China and Lehrstuhl C für Mathematik, RWTH Aachen University, 52056 Aachen, Germany;School of Mathematical Science, Shanxi University, 030006 Taiyuan, PR China;Lehrstuhl C für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

For a positive integer k, a graph G is k-ordered if for every ordered set of k vertices, there is a cycle that encounters the vertices of the set in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonian. We first show that if G is a (k+1)-connected, k-ordered graph of order n=4k+3 and d(u)+d(v)=n-1 for every pair of vertices u and v of G with d(u,v)=2, then G is k-ordered Hamiltonian unless G belongs to an exceptional class of graphs. The latter class is described in this paper. By this result, we prove that G is k-ordered Hamiltonian if G has the order n=27k^3 and d(u)+d(v)=n+(3k-9)/2 for every pair of vertices u and v of G with d(u,v)=2.